Domination Critical Knodel Graphs

TL;DR

This paper investigates the domination criticality and stability of Kn"odel graphs, characterizing 3-regular and 4-regular cases in terms of their domination number properties.

Contribution

It provides a characterization of the domination criticality and stability of Kn"odel graphs, focusing on 3-regular and 4-regular cases, which was previously unexplored.

Findings

Characterized 3-regular Kn"odel graphs as $oldsymbol{eta}$-critical or $oldsymbol{eta}$-stable.

Characterized 4-regular Kn"odel graphs as $oldsymbol{eta}$-critical or $oldsymbol{eta}$-stable.

Established conditions for domination criticality and stability in these graphs.

Abstract

A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is called domination vertex critical, or just $\gamma$-critical if removal of any vertex decreases the domination number. A graph $G$ is called domination vertex stable, or just $\gamma$-stable, if removal of any vertex does not decrease the domination number. For an even integer $n\ge 2$ and $1\le \Delta \le \lfloor \log_2n \rfloor$, a Kn\"odel graph $W_{\Delta,n}$ is a $\Delta$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j \le n/2-1$, where for every $j$, $0\le j \le n/2-1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j+2^k-1$ (mod (n/2)), for $k=0,1,\cdots,\Delta-1$. in this paper, we study the domination criticality and domination stability of Kn\"odel graphs. We charactrize the 3-regular and 4-regular Kn\"odel graphs by $\gamma$-criticality or $\gamma$-stability.